Subgroups — #10948320

Order 768.

520 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2875	C₂	1	1	yes	maximal, characteristic
2	#72007	C₂	1	2	yes	maximal
3	#69886	C₂	1	4	yes	maximal, complemented
4	#69885	C₂	1	2	yes	maximal, complemented
5	#69864	C₂	1	4	yes	maximal, complemented
6	C₂×#4466	C₂	1	2	yes	maximal, complemented
7	#33426	—	3	1	no	maximal, Sylow
8	C₂×C₃×#131	C₄	1	1	yes	characteristic
9	C₂×C₃×#131	C₂²	1	1	yes	characteristic
10	C₃×C₄×#41	C₄	1	1	yes	characteristic
11	C₃×#405	C₄	1	2	yes	—
12	C₃×#405	C₄	1	1	yes	characteristic
13	C₃×#405	C₂²	1	2	yes	—
14	C₃×#405	C₂²	1	1	yes	characteristic
15	C₃×C₄×M₁₆	C₄	1	1	yes	characteristic
16	C₃×C₄×M₁₆	C₂²	1	2	yes	—
17	C₂×C₃×#97	C₄	1	2	yes	—
18	C₂×C₃×#97	C₂²	1	1	yes	characteristic
19	C₂×#812	C₂²	1	4	yes	—
20	#4553	C₂²	1	8	yes	—
21	#4466	C₂²	1	4	yes	—
22	#4466	C₂²	1	4	yes	—
23	#4463	C₂²	1	8	yes	—
24	#2875	D₃	1	1	yes	characteristic
25	#2166	—	3	4	no	—
26	#2165	—	3	4	no	—
27	#2153	—	3	2	no	—
28	#2115	—	3	2	no	—
29	C₂×#347	—	3	2	no	—
30	C₂×C₃×#41	C₂×C₄	1	1	yes	characteristic
31	C₃×#131	—	2	4	no	—
32	C₃×#131	—	2	4	no	—
33	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×M₁₆	C₂³	1	2	yes	—
37	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
38	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
40	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
41	C₃×C₄×Q₈	—	2	1	no	—
42	C₃×C₄×D₄	—	2	2	no	—
43	C₃×C₄×D₄	—	2	1	no	—
44	C₃×#117	C₂×C₄	1	2	yes	—
45	C₃×#117	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
47	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
48	C₃×#97	C₂×C₄	1	1	yes	characteristic
49	C₃×#97	C₂×C₄	1	2	yes	—
50	C₃×#97	C₂×C₄	1	2	yes	—
51	C₃×#97	C₂×C₄	1	2	yes	—
52	C₃×#97	C₂×C₄	1	1	yes	characteristic
53	C₃×#97	C₂³	1	1	yes	characteristic
54	C₃×#97	C₂³	1	1	yes	characteristic
55	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
56	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
57	C₃×C₄×C₈	C₂³	1	2	yes	—
58	#812	—	2	4	no	—
59	#812	C₂³	1	8	yes	—
60	C₂×C₃⋊C₁₆	—	2	4	no	—
61	C₂×#131	Dic₃	1	1	yes	characteristic
62	C₂×#131	C₂×D₃	1	1	yes	characteristic
63	C₄×#41	Dic₃	1	1	yes	characteristic
64	C₂×M₃₂	—	3	4	no	—
65	#405	Dic₃	1	2	yes	—
66	#405	Dic₃	1	1	yes	characteristic
67	#405	C₂×D₃	1	2	yes	—
68	#405	C₂×D₃	1	1	yes	characteristic
69	C₄×M₁₆	Dic₃	1	1	yes	characteristic
70	C₄×M₁₆	C₂×D₃	1	2	yes	—
71	C₂×#97	Dic₃	1	2	yes	—
72	C₂×#97	C₂×D₃	1	1	yes	characteristic
73	#364	—	3	8	no	—
74	#349	—	3	8	no	—
75	#347	—	3	4	no	—
76	#347	—	3	4	no	—
77	C₃×#41	—	2	2	no	—
78	C₃×#41	—	2	2	no	—
79	C₂×C₃×Q₈	M₁₆	1	1	yes	characteristic
80	C₂×C₃×D₄	M₁₆	1	2	yes	—
81	C₂×C₃×D₄	M₁₆	1	1	yes	characteristic
82	C₂²×C₃×C₄	—	2	2	no	—
83	C₂²×C₃×C₄	—	2	1	no	—
84	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
85	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
86	C₃×M₁₆	—	2	2	no	—
87	C₃×M₁₆	—	2	4	no	—
88	C₃×M₁₆	—	2	4	no	—
89	C₃×M₁₆	M₁₆	1	4	yes	—
90	C₂×C₃×C₈	—	2	2	no	—
91	C₂×C₃×C₈	—	2	4	no	—
92	C₂×C₃×C₈	—	2	4	no	—
93	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
98	C₂×C₃×C₈	#41	1	4	yes	—
99	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
100	C₃×C₄⋊C₄	—	2	2	no	—
101	C₃×C₄⋊C₄	—	2	1	no	—
102	C₃×C₂²⋊C₄	—	2	2	no	—
103	C₃×C₂²⋊C₄	—	2	1	no	—
104	C₃×C₄²	—	2	2	no	—
105	C₃×C₄²	—	2	1	no	—
106	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
107	C₃×C₄²	C₂²×C₄	1	2	yes	—
108	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
109	C₃⋊C₁₆	—	2	8	no	—
110	C₂×#41	C₂×Dic₃	1	1	yes	characteristic
111	#131	—	2	4	no	—
112	#131	—	2	4	no	—
113	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
114	C₂×M₁₆	C₂×Dic₃	1	1	yes	characteristic
115	C₂×M₁₆	C₂×Dic₃	1	1	yes	characteristic
116	C₂×M₁₆	C₂²×D₃	1	2	yes	—
117	C₂²×C₈	C₂×Dic₃	1	2	yes	—
118	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
119	C₂²×C₈	C₂×Dic₃	1	2	yes	—
120	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
121	C₄×Q₈	—	2	1	no	—
122	C₄×D₄	—	2	2	no	—
123	C₄×D₄	—	2	1	no	—
124	#117	C₂×Dic₃	1	2	yes	—
125	#117	C₂×Dic₃	1	1	yes	characteristic
126	C₂×C₄²	C₂×Dic₃	1	2	yes	—
127	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
128	M₃₂	—	3	8	no	—
129	M₃₂	—	6	4	no	—
130	C₂×C₁₆	—	6	4	no	—
131	#97	C₂×Dic₃	1	1	yes	characteristic
132	#97	C₂×Dic₃	1	2	yes	—
133	#97	C₂×Dic₃	1	2	yes	—
134	#97	C₂×Dic₃	1	2	yes	—
135	#97	C₂×Dic₃	1	1	yes	characteristic
136	#97	C₂²×D₃	1	1	yes	characteristic
137	#97	C₂²×D₃	1	1	yes	characteristic
138	C₄×C₈	C₂×Dic₃	1	2	yes	—
139	C₄×C₈	C₂×Dic₃	1	2	yes	—
140	C₄×C₈	C₂²×D₃	1	2	yes	—
141	C₂³×C₃	C₂×M₁₆	1	2	yes	—
142	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
143	C₃×Q₈	—	2	2	no	—
144	C₃×D₄	—	2	4	no	—
145	C₃×D₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	8	1	no	—
152	C₂×C₃×C₄	—	8	2	no	—
153	C₂×C₃×C₄	#117	1	2	yes	—
154	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
155	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
156	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
158	C₃×C₈	—	2	2	no	—
159	C₃×C₈	—	2	1	no	—
160	C₃×C₈	—	2	1	no	—
161	C₃×C₈	#117	1	2	yes	—
162	C₃×C₈	C₂×M₁₆	1	4	yes	—
163	C₃×C₈	C₂×#41	1	2	yes	—
164	#41	—	2	2	no	—
165	#41	—	2	2	no	—
166	C₂×Q₈	#208	1	1	yes	characteristic
167	C₂×D₄	#208	1	2	yes	—
168	C₂×D₄	#208	1	1	yes	characteristic
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	C₂²×Dic₃	1	2	yes	—
172	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
173	M₁₆	—	2	2	no	—
174	M₁₆	—	2	4	no	—
175	M₁₆	—	2	4	no	—
176	M₁₆	#208	1	4	yes	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	#235	1	4	yes	—
181	C₂×C₈	C₂²×Dic₃	1	2	yes	—
182	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
183	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
184	C₂×C₈	C₂²×Dic₃	1	2	yes	—
185	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
186	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	1	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	2	1	no	—
193	C₄²	C₂²×Dic₃	1	1	yes	characteristic
194	C₄²	C₂²×Dic₃	1	2	yes	—
195	C₄²	C₂²×Dic₃	1	1	yes	characteristic
196	C₁₆	—	6	8	no	—
197	C₂²×C₃	—	2	2	no	—
198	C₂²×C₃	—	4	1	no	—
199	C₂²×C₃	—	4	2	no	—
200	C₂²×C₃	#432	1	2	yes	—
201	C₂²×C₃	C₂²×M₁₆	1	1	yes	characteristic
202	C₃×C₄	—	2	2	no	—
203	C₃×C₄	—	4	1	no	—
204	C₃×C₄	—	4	2	no	—
205	C₃×C₄	—	4	1	no	—
206	C₃×C₄	#432	1	2	yes	—
207	C₃×C₄	C₂×#117	1	1	yes	characteristic
208	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
209	C₂³	C₂×#208	1	2	yes	—
210	C₂³	C₂×#208	1	1	yes	characteristic
211	Q₈	—	2	2	no	—
212	D₄	—	2	4	no	—
213	D₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	8	1	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	C₂×#208	1	2	yes	—
222	C₂×C₄	C₂×#208	1	1	yes	characteristic
223	C₂×C₄	C₂×#208	1	2	yes	—
224	C₂×C₄	#926	1	2	yes	—
225	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
226	C₈	—	2	1	no	—
227	C₈	—	2	1	no	—
228	C₈	—	2	2	no	—
229	C₈	C₂×#208	1	4	yes	—
230	C₈	#926	1	2	yes	—
231	C₈	C₂×#235	1	2	yes	—
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	—	2	2	no	—
234	C₂×C₃	—	4	2	no	—
235	C₂×C₃	C₂×#432	1	1	yes	characteristic
236	C₂²	—	2	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	2	no	—
239	C₂²	#4926	1	2	yes	—
240	C₂²	C₂²×#208	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	2	no	—
245	C₄	#4926	1	2	yes	—
246	C₄	C₂²×#208	1	1	yes	characteristic
247	C₄	C₂×#926	1	1	yes	characteristic
248	C₃	#33426	1	1	yes	characteristic, complemented, Sylow
249	C₂	—	2	1	no	—
250	C₂	—	2	2	no	—
251	C₂	—	4	2	no	—
252	C₂	C₂×#4926	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#926
commutator	C₂×C₃×C₄	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₃×#2875	C₂
socle	C₂×C₃	C₂×#432
prime radical	C₂×C₃×#41	C₂×C₄
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2875	D₃
hypocenter	C₃	#33426
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×#97 ◁ #2875

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #69886	C₂	semidirect
4. #69885	C₂	semidirect
5. #69864	C₂	semidirect
6. C₂×#4466	C₂	semidirect
248. C₃	#33426	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#2875	2
#72007	2
#69886	2
#69885	2
#69864	2
C₂×#4466	2
#33426	3